In this paper, we shall develop a generic channel estimation framework based on the convex formulation for dense cloud radio access networks (Cloud-RAN). Due to the training resource constraint and the large number of transmit antennas, the pilot length is smaller than the antenna number, and thus channel estimation becomes an ill-posed inverse problem. By observing that the wireless channel possesses ample exploitable statistical characteristics, we propose to convert the available spatial and temporal prior information into appropriate convex regularizing functions, yielding convex optimization formulations for the underdetermined channel estimation problem. Simulation results demonstrate that exploiting the prior information of large-scale fading and temporal correlation can achieve good estimation performance even with limited training resources. The alternating direction method of multipliers (ADMM) algorithm is further adopted to solve the resultant large-scale channel estimation problems. The proposed framework is, therefore, scalable to the overhead of prior information and the computation cost for large network sizes.